1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Airy Ai' Function</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../airy.html" title="Airy Functions"> 9<link rel="prev" href="bi.html" title="Airy Bi Function"> 10<link rel="next" href="bip.html" title="Airy Bi' Function"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="bi.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../airy.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="bip.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="math_toolkit.airy.aip"></a><a class="link" href="aip.html" title="Airy Ai' Function">Airy Ai' Function</a> 28</h3></div></div></div> 29<h5> 30<a name="math_toolkit.airy.aip.h0"></a> 31 <span class="phrase"><a name="math_toolkit.airy.aip.synopsis"></a></span><a class="link" href="aip.html#math_toolkit.airy.aip.synopsis">Synopsis</a> 32 </h5> 33<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">airy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 34</pre> 35<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> 36 37 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 38 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">airy_ai_prime</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> 39 40 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span> 41 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">airy_ai_prime</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&);</span> 42 43<span class="special">}}</span> <span class="comment">// namespaces</span> 44</pre> 45<h5> 46<a name="math_toolkit.airy.aip.h1"></a> 47 <span class="phrase"><a name="math_toolkit.airy.aip.description"></a></span><a class="link" href="aip.html#math_toolkit.airy.aip.description">Description</a> 48 </h5> 49<p> 50 The function <a class="link" href="aip.html" title="Airy Ai' Function">airy_ai_prime</a> calculates 51 the Airy function Ai' which is the derivative of the first solution to the 52 differential equation: 53 </p> 54<div class="blockquote"><blockquote class="blockquote"><p> 55 <span class="inlinemediaobject"><img src="../../../equations/airy.svg"></span> 56 57 </p></blockquote></div> 58<p> 59 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 60 be used to control the behaviour of the function: how it handles errors, 61 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 62 documentation for more details</a>. 63 </p> 64<p> 65 The following graph illustrates how this function changes as <span class="emphasis"><em>x</em></span> 66 changes: for negative <span class="emphasis"><em>x</em></span> the function is cyclic, while 67 for positive <span class="emphasis"><em>x</em></span> the value tends to zero: 68 </p> 69<div class="blockquote"><blockquote class="blockquote"><p> 70 <span class="inlinemediaobject"><img src="../../../graphs/airy_aip.svg" align="middle"></span> 71 72 </p></blockquote></div> 73<h5> 74<a name="math_toolkit.airy.aip.h2"></a> 75 <span class="phrase"><a name="math_toolkit.airy.aip.accuracy"></a></span><a class="link" href="aip.html#math_toolkit.airy.aip.accuracy">Accuracy</a> 76 </h5> 77<p> 78 This function is implemented entirely in terms of the Bessel functions <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a> and <a class="link" href="../bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</a> - refer to those 79 functions for detailed accuracy information. 80 </p> 81<p> 82 In general though, the relative error is low (less than 100 ε) for <span class="emphasis"><em>x 83 > 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x < 84 0</em></span> as the following error plot illustrates: 85 </p> 86<div class="blockquote"><blockquote class="blockquote"><p> 87 <span class="inlinemediaobject"><img src="../../../graphs/ai_prime__double.svg" align="middle"></span> 88 89 </p></blockquote></div> 90<h5> 91<a name="math_toolkit.airy.aip.h3"></a> 92 <span class="phrase"><a name="math_toolkit.airy.aip.testing"></a></span><a class="link" href="aip.html#math_toolkit.airy.aip.testing">Testing</a> 93 </h5> 94<p> 95 Since this function is implemented in terms of other special functions, there 96 are only a few basic sanity checks, using test values from <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>. 97 </p> 98<h5> 99<a name="math_toolkit.airy.aip.h4"></a> 100 <span class="phrase"><a name="math_toolkit.airy.aip.implementation"></a></span><a class="link" href="aip.html#math_toolkit.airy.aip.implementation">Implementation</a> 101 </h5> 102<p> 103 This function is implemented in terms of the Bessel functions using the relations: 104 </p> 105<div class="blockquote"><blockquote class="blockquote"><p> 106 <span class="inlinemediaobject"><img src="../../../equations/airy_aip.svg"></span> 107 108 </p></blockquote></div> 109</div> 110<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 111<td align="left"></td> 112<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 113 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 114 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 115 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 116 Daryle Walker and Xiaogang Zhang<p> 117 Distributed under the Boost Software License, Version 1.0. (See accompanying 118 file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) 119 </p> 120</div></td> 121</tr></table> 122<hr> 123<div class="spirit-nav"> 124<a accesskey="p" href="bi.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../airy.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="bip.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 125</div> 126</body> 127</html> 128